Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nlin/0510048 · International Journal of Mathematics and Mathematical Sciences

Quantum curve in q‐oscillator model

С. М. Сергеев

Published Jan 1, 2006• 18 citations
arXivHTMLPDF

Abstract

A lattice model of interacting q‐oscillators, proposed by V. Bazhanov and S. Sergeev in 2005 is the quantum‐mechanical integrable model in 2 + 1 dimensional space‐time. Its layer‐to‐layer transfer matrix is a polynomial of two spectral parameters, it may be regarded in terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of length M and as a sum of sl(M) transfer matrices of a chain of length N for reducible representations. The aim of this paper is to derive the Bethe ansatz equations for the q‐oscillator model entirely in the framework of 2 + 1 integrability providing the evident rank‐size duality.

DOI
10.1155/ijmms/2006/92064

Related papers

Ansatz of Hans Bethe for a two-dimensional lattice Bose gas

С. М. Сергеев

The method of q -oscillator lattices, proposed recently in Bazhanov and Sergeev 2005 ( Preprint hep-th/0509181), provides the tool for a construction of various integrable models of quantum mechanics in (2 + 1)-dimensional spacetime. In contrast to any one-dimensional quantum chain, its two-dimensional generalizations—quantum lattices—admit different geometrical structures. In this paper, we consider the q -oscillator model on a special lattice. The model may be interpreted as a two-dimensional lattice Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe ansatz: the p -particles' wavefunction is the sum of plane waves. Consistency conditions is the set of 2 p equations for p one-particle wave vectors. These 'Bethe ansatz' equations are the main result of this paper.

arXiv:nlin/0511043•Journal of Physics A Mathematical and General•Mar 8, 2006•4 citations

Baxter's T-Q Relation and Bethe Ansatz of Discrete Quantum Pendulum and Sine-Gordon Model

Shao-shiung Lin, Shi-shyr Roan

Using the Baxter's T-Q relation derived from the transfer matrix technique, we consider the diagonalization problem of discrete quantum pendulum and discrete quantum sine-Gordon Hamiltonian from the algebraic geometry aspect. For a finite chain system of the size L, when the spectral curve degenerates into rational curves, we have reduced Baxter's T-Q relation into a polynomial equation; the connection of T-Q polynomial equation with the algebraic Bethe Ansatz is clearly established . In particular, for L=4 it is the case of rational spectral curves for the discrete quantum pendulum and discrete sine-Gordon model. To these Baxter's T-Q polynomial equations, we have obtained the complete and explicit solutions with a detailed understanding of their quantitative and qualitative structure. In general the model possesses a spectral curve with a generic parameter. We have conducted certain qualitative study on the algebraic geometry of this high-genus Riemann surface incorporating Baxter's T-Q relation.

arXiv:hep-th/0105140•May 15, 2001•1 citation

Quantum Groups and Integrable Models

P. P. Kulish

The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Birkhäuser Basel eBooks•Jan 1, 2003•1 citation

The Bethe ansatz for a new integrable open quantum system

Marius de Leeuw, Chiara Paletta

In this paper we apply the nested algebraic Bethe ansatz to compute the eigenvalues and the Bethe equations of the transfer matrix of the new integrable Lindbladian found in [1]. We show that it can be written as an integrable spin chain consisting of two interacting XXZ spin chains. We numerically compute the Liouville gap and its dependence on the parameters in the system such as scaling with the system length and interaction strength.

arXiv:2207.14193•Jul 28, 2022•2 citations

Integrability in three dimensions: Algebraic Bethe ansatz for anyonic models

Sh. Khachatryan, A. Ferraz, Andreas Klümper +1 more

We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.

Nuclear Physics B•Aug 13, 2015•5 citations

Bethe ansatz for the three-layer Zamolodchikov model

Herman E. Boos, Vladimir V. Mangazeev

This paper is a continuation of our previous work (Boos H E and Mangazeev V V 1999 J. Phys. A: Math. Gen. 32 3041-54). We obtain two more functional relations for the eigenvalues of the transfer matrices for the sl (3) chiral Potts model at q 2 = -1. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.

arXiv:solv-int/9903010•Journal of Physics A Mathematical and General•Jan 1, 1999•6 citations
A lattice model of interacting q‐oscillators, proposed by V. Bazhanov and S. Sergeev in 2005 is the quantum‐mechanical integrable model in 2 + 1 dimensional space‐time. Its layer‐to‐layer transfer matrix is a polynomial of two spectral parameters, it may be regarded in terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of length M and as a sum of sl(M) transfer matrices of a chain of length N for reducible representations. The aim of this paper is to derive the Bethe ansatz equations for the q‐oscillator model entirely in the framework of 2 + 1 integrability providing the evident rank‐size duality.
The method of q -oscillator lattices, proposed recently in Bazhanov and Sergeev 2005 ( Preprint hep-th/0509181), provides the tool for a construction of various integrable models of quantum mechanics in (2 + 1)-dimensional spacetime. In contrast to any one-dimensional quantum chain, its two-dimensional generalizations—quantum lattices—admit different geometrical structures. In this paper, we consider the q -oscillator model on a special lattice. The model may be interpreted as a two-dimensional lattice Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe ansatz: the p -particles' wavefunction is the sum of plane waves. Consistency conditions is the set of 2 p equations for p one-particle wave vectors. These 'Bethe ansatz' equations are the main result of this paper.
Using the Baxter's T-Q relation derived from the transfer matrix technique, we consider the diagonalization problem of discrete quantum pendulum and discrete quantum sine-Gordon Hamiltonian from the algebraic geometry aspect. For a finite chain system of the size L, when the spectral curve degenerates into rational curves, we have reduced Baxter's T-Q relation into a polynomial equation; the connection of T-Q polynomial equation with the algebraic Bethe Ansatz is clearly established . In particular, for L=4 it is the case of rational spectral curves for the discrete quantum pendulum and discrete sine-Gordon model. To these Baxter's T-Q polynomial equations, we have obtained the complete and explicit solutions with a detailed understanding of their quantitative and qualitative structure. In general the model possesses a spectral curve with a generic parameter. We have conducted certain qualitative study on the algebraic geometry of this high-genus Riemann surface incorporating Baxter's T-Q relation.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.